The most common theoretical approach to studying the correlated states is to start with a continuum effective Hamiltonian, often referred to as the Bistritzer-MacDonald (BM) model [1]}, which gives isolated narrow bands for a range of near-magic twist angles, and then to project the Coulomb interaction onto the wavefunctions of the narrow bands (sometimes including few remote bands as well).
The BM model [1]} has achieved success in many respects. It correctly predicts the first magic angle where the bands around the charge neutrality point (CNP) become extremely narrow and captures their band topology.
For relaxed structures, however, the BM model –which was originally derived for a rigid twist– does not include terms which are nominally of the same order in gradient expansion as the ones which are kept, such as the pseudo-magnetic fields induced by the \(C_3\)  symmetric strain from lattice relaxation. Moreover, next order gradient terms are needed to accurately capture the narrow bands near the magic angle due to the anomalously small non-interacting bandwidth obtained without such terms [3]}. In addition, the narrow band wavefunctions of the BM model are nearly particle-hole (p-h) symmetric [4]}. The presence of the p-h symmetry is known to play important role in choosing the correlated ground states [5]}, [6]}, [4]}, [8]}. Experimentally, it is also seen to be broken at low temperature in that various correlated states appear more stable on either the hole or the electron side of the CNP. This motivates development of a more accurate low energy effective continuum model for TBG.
